Hydrodynamics of Micro-swimmers in Films

TL;DR

This paper develops a mathematical framework to analyze how micro-swimmers behave in viscous films, revealing how surfaces influence their flow fields and trajectories, with implications for understanding microbial interactions near interfaces.

Contribution

It introduces a recursive image system approximation for swimmer flow fields in films and extends existing solutions to new geometries, providing analytical tools for studying micro-swimmer dynamics.

Findings

Swimmer accumulation is biased towards the no-slip wall in thick films.

Higher-order multipoles can counteract the dipole-induced bias.

Proposed experimental method to determine swimmer position via circular trajectories.

Abstract

One of the principal mechanisms by which surfaces and interfaces affect microbial life is by perturbing the hydrodynamic flows generated by swimming. By summing a recursive series of image systems we derive a numerically tractable approximation to the three-dimensional flow fields of a Stokeslet (point force) within a viscous film between a parallel no-slip surface and no-shear interface and, from this Green's function, we compute the flows produced by a force- and torque-free micro-swimmer. We also extend the exact solution of Liron & Mochon (1976) to the film geometry, which demonstrates that the image series gives a satisfactory approximation to the swimmer flow fields if the film is sufficiently thick compared to the swimmer size, and we derive the swimmer flows in the thin-film limit. Concentrating on the thick film case, we find that the dipole moment induces a bias towards swimmer accumulation at the no-slip wall rather than the water-air interface, but that higher-order multipole moments can oppose this. Based on the analytic predictions we propose an experimental method to find the multipole coefficient that induces circular swimming trajectories, allowing one to analytically determine the swimmer's three-dimensional position under a microscope.